allowed us to compute the second Rényi entro-
py. We used an iterative Bayesian scheme ( 32 )
to mitigate measurement errors and remove
undersampling bias [we also compare random-
ized measurement with tomography results in
( 24 ), section III].
Distributions of the measured entanglement
entropies are shown in Fig. 2C for subsystems
of 2 by 2, 2 by 3, and 3 by 3 qubits within the
toric code ground state. For a subsystem with
nqubits, the entanglement entropy ranges
from 0 for a product state tonln2. In the toric
code, subsystems with no interior have the
maximum valuenln2; in those cases, we mea-
sured a narrow distribution centered just below
the ideal value. For subsystems with an inte-
rior, we measured a wider distribution cen-
tered slightly above the predicted value. This is
consistent with unitary error and decoherence
slightly mixing the system with its environ-
ment, which increases entanglement entro-
pies that are not yet at their maximal value.
We computedStopofrom the subsystem
entropies using Eq. 3 for 14 different 2-by-2
arrays, 20 different 2-by-3 arrays, and three
different 3-by-3 arrays. Each randomized mea-
surement on the qubit array yields several
Stopoestimates from different orientations
of the partitionsA,B, andC. Distributions
of measuredStopoare shown in Fig. 2D, with
mean valuesStopo/ln2 =–0.89,–0.90, and–0.95
for the 2-by-2, 2-by-3, and 3-by-3 qubit arrays,
respectively. The distributions provide strong
evidence for the nontrivial topological nature
of the state, approaching the ideal value of
Stopo=–ln2, which is incompatible with the
trivial state value of zero.
Simulation of braiding statistics
Our approach of directly preparing the toric
code ground state also allows us to simulate
the exotic braiding statistics of its quasipar-
ticle excitations (anyons). We used a mapping
between the adiabatic evolution of toric code
excitations and strings of Pauli operators
applied to the ground state. Within this
framework, a controlled Pauli string imple-
ments an interferometry protocol, through
which we experimentally extracted the mutual
and exchange statistics corresponding to all
combinations of excitations.
The quasiparticle excitations of the toric
code are commonly denoted as“electric”e
withh i¼ As 1, and“magnetic”mwithh i¼Bp
1, in connection to lattice gauge theory. The
four distinct anyons of the toric code are 1 (the
absence of aneorm),e,m, andy(an emergent
fermion resulting from the combination ofe
andm). In the toric code, the mutual statistics
are encoded in the phase accumulated when
dragging one anyon around another anyon of
a different type, and the exchange statistics are
phases that arise from the spatial interchange
of two identical anyons. The toric code excited
states can be created by applying a string of
Pauli operators to the ground state: AnX-string
will result in the state witheexcitations at each
end, whereas aZ-string prepares the state with
mexcitations at each end. We visualize an
example ofe–mmutual braiding in Fig. 3A,
with snapshots of experimentally measured
parity values,hiAs andhiBp.Wemovedane
aroundmwith anX-string, eventually re-
turning to its initial position. The initial and
final states have the same parity values but
differ by an overall phase—in this case,p, which
is not directly detectable.
To experimentally determine this phase, we
used multiqubit Ramsey interferometry ( 33 ).
This protocol provides a scalable way to mea-
sure the overlap between the initial and final
states, allowing experimental access to the
accumulated phaseq. A key step in this pro-
tocol is the use of an auxiliary qubit and a
controlled operation, effectively creating a
superposition of the braided and nonbraided
states (Fig. 3, B and C). This sequence imparts
qinto a measurable rotation of the auxiliary
qubit. We efficiently compiled the multiqubit
controlled operations into CZ gates. Because
the measured phases are sensitive to coherent
and non-Markovian errors, we used random-ized compiling to mitigate these errors ( 34 , 35 ).
Details are available in ( 24 ), section IV.
Illustrated in Fig. 3, D and E, are two ex-
amples of braiding interferometry by use of
minimal-length paths, which we chose because
of the zero correlation length of the states. In
Fig. 3D, we extract thee–mmutual statistics,
where the braiding path is a Pauli stringXXXX,
movingearound the plaquette that contains
anm. A similar example is shown in Fig. 3E
for the exchange statistics of two identicaly
excitations by using a path of intertwining
Pauli strings ofXXXXandZZZZ, simplifying
toXXYYZZ[( 24 ), section VI]. The parity mea-
surements show consistent values before and
after the controlled-braiding operation, slightly
fading owing to decoherence and gate error.
We measured the phases for the other mutual
and exchange combinations and provide the
results in Fig. 3F, in which the phases are
plotted alongside their corresponding braid
diagrams, with the expected values 0 andp
indicated with dashed gray lines.
Our measurements illuminate the nontrivial
mutual and exchange statistics of the toric
code. Braidingearoundmresults in apphase,
which does not occur for local bosons or fer-
mions. Moreover, althougheandmboth satisfySCIENCEscience.org 3 DECEMBER 2021•VOL 374 ISSUE 6572 1239
Fig. 2. Topological entanglement entropy.(A) Illustration of subsystemsA,B, andCused to measure
topological entanglement entropyStopoon four-, six-, and nine-qubit systems within the toric code lattice.
(B) Expected entanglement entropySfor groups of qubits in the toric code. We drew a red perimeter
around each group and counted the numberkof star operators (blue tiles) that it crosses:S=kln2 +Stopo=
(k–1)ln2. (C) Second Rényi entropyS(2)distributions measured on the 31-qubit toric code ground
state. A histogram is shown for each subsystem shape. Dashed gray lines indicate the predicted integer
values forjiG.(D) Topological entanglement entropyStopo/ln2 (ideal value–1) computed from the entropies
in (C). We evaluated each dataset in all possible orientations of the subsystems in (A) (2 by 2, 4; 2 by 3,
2; 3 by 3, 8). In (D), top to bottom, the numbers at top right show the mean (also indicated by the
dark green line) and the standard deviation of the distribution.RESEARCH | RESEARCH ARTICLES